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2015 Large BMO spaces vs interpolation
Jose Conde-Alonso, Tao Mei, Javier Parcet
Anal. PDE 8(3): 713-746 (2015). DOI: 10.2140/apde.2015.8.713

Abstract

We introduce a class of BMO spaces which interpolate with Lp and are sufficiently large to serve as endpoints for new singular integral operators. More precisely, let (Ω,Σ,μ) be a σ-finite measure space. Consider two filtrations of Σ by successive refinement of two atomic σ-algebras Σa and Σb having trivial intersection. Construct the corresponding truncated martingale BMO spaces. Then, the intersection seminorm only leaves out constants and we provide a quite flexible condition on (Σa,Σb) so that the resulting space interpolates with Lp in the expected way. In the presence of a metric d, we obtain endpoint estimates for Calderón–Zygmund operators on (Ω,μ,d) under additional conditions on (Σa,Σb). These are weak forms of the “isoperimetric” and the “locally doubling” properties of Carbonaro, Mauceri and Meda which admit less concentration at the boundary. Examples of particular interest include densities of the form e±|x|α for any α > 0 or (1 + |x|β)1 for any β n32. A (limited) comparison with Tolsa’s RBMO is also possible. On the other hand, a more intrinsic formulation yields a Calderón–Zygmund theory adapted to regular filtrations over (Σa,Σb) without using a metric. This generalizes well-known estimates for perfect dyadic and Haar shift operators. In contrast to previous approaches, ours extends to matrix-valued functions (via recent results from noncommutative martingale theory) for which only limited results are known and no satisfactory nondoubling theory exists so far.

Citation

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Jose Conde-Alonso. Tao Mei. Javier Parcet. "Large BMO spaces vs interpolation." Anal. PDE 8 (3) 713 - 746, 2015. https://doi.org/10.2140/apde.2015.8.713

Information

Received: 9 July 2014; Revised: 18 January 2015; Accepted: 6 March 2015; Published: 2015
First available in Project Euclid: 16 November 2017

zbMATH: 1316.42027
MathSciNet: MR3353829
Digital Object Identifier: 10.2140/apde.2015.8.713

Subjects:
Primary: 42B20, 42B35, 46L52, 60G46

Rights: Copyright © 2015 Mathematical Sciences Publishers

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